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Pries: 676 Number Theory. 2010. Homework 8. More about ramification 1. Let K = C(x) and L = K[y, z]/(y 2 − πx, z 3 − (y + √ π)/(y − √ π)) A. Show that Gal(L : K) = S3 . B. Describe the primes of C[x] which are ramified in L. 2. If L|K is a Galois extension of number fields whose Galois group is non-cyclic, show there are at most finitely many prime ideals for K which are non-split in L. 3. Suppose L|Q is a Galois extension. Suppose Q is a prime ideal of B = OL which divides the prime ideal p of Z. Suppose p does not ramify in B. Show there is exactly one automorphism σQ ∈ Gal(L : K) so that σ(b) ≡ bp mod Q for all b ∈ B. We call σQ the Frobenius automorphism. Show that GQ is cyclic and generated by σQ . 4. Choose a topic for a 15 minute presentation. I’m happy to help brainstorm topics.